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Theorem abrexco 5463
 Description: Composition of two image maps C(y) and B(w). (Contributed by set.mm contributors, 27-May-2013.)
Hypotheses
Ref Expression
abrexco.1 B V
abrexco.2 (y = BC = D)
Assertion
Ref Expression
abrexco {x y {z w A z = B}x = C} = {x w A x = D}
Distinct variable groups:   y,A,z   y,B,z   w,C   y,D   x,w,y   z,w
Allowed substitution hints:   A(x,w)   B(x,w)   C(x,y,z)   D(x,z,w)

Proof of Theorem abrexco
StepHypRef Expression
1 df-rex 2620 . . . 4 (y {z w A z = B}x = Cy(y {z w A z = B} x = C))
2 vex 2862 . . . . . . . 8 y V
3 eqeq1 2359 . . . . . . . . 9 (z = y → (z = By = B))
43rexbidv 2635 . . . . . . . 8 (z = y → (w A z = Bw A y = B))
52, 4elab 2985 . . . . . . 7 (y {z w A z = B} ↔ w A y = B)
65anbi1i 676 . . . . . 6 ((y {z w A z = B} x = C) ↔ (w A y = B x = C))
7 r19.41v 2764 . . . . . 6 (w A (y = B x = C) ↔ (w A y = B x = C))
86, 7bitr4i 243 . . . . 5 ((y {z w A z = B} x = C) ↔ w A (y = B x = C))
98exbii 1582 . . . 4 (y(y {z w A z = B} x = C) ↔ yw A (y = B x = C))
101, 9bitri 240 . . 3 (y {z w A z = B}x = Cyw A (y = B x = C))
11 rexcom4 2878 . . 3 (w A y(y = B x = C) ↔ yw A (y = B x = C))
12 abrexco.1 . . . . 5 B V
13 abrexco.2 . . . . . 6 (y = BC = D)
1413eqeq2d 2364 . . . . 5 (y = B → (x = Cx = D))
1512, 14ceqsexv 2894 . . . 4 (y(y = B x = C) ↔ x = D)
1615rexbii 2639 . . 3 (w A y(y = B x = C) ↔ w A x = D)
1710, 11, 163bitr2i 264 . 2 (y {z w A z = B}x = Cw A x = D)
1817abbii 2465 1 {x y {z w A z = B}x = C} = {x w A x = D}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by: (None)
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